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Base-40 arithmetic for music apps

Base-40 arithmetic for music apps. Music 253/CS 275A Stanford University. Where did Base-40 come from?. Conceived by Walter Hewlett (1986 ); first pub 1992 Goals: enharmonic spelling preservation, correct analysis, correct transposition

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Base-40 arithmetic for music apps

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  1. Base-40 arithmetic for music apps Music 253/CS 275A Stanford University

  2. Where did Base-40 come from? • Conceived by Walter Hewlett (1986); first pub 1992 • Goals: enharmonic spelling preservation, correct analysis, correct transposition • Reproduced at http://www.ccarh.org/publications/reprints/ • Further elaborated in U.S. Patent 5,675,100 (7 October 1997) http://www.google.com/patents/US5675100 2013 Eleanor Selfridge-Field

  3. Why Base-40? • Musical literacy • Tonal legibility (common-practice era) • Musical computation in integer arithmetic • Intervallic complementarity Base-10 complementarity: If interval = 3, complement = 7 If interval = 6, complement = 4 2013 Eleanor Selfridge-Field

  4. Review: Interval sizes and qualities 2013 Eleanor Selfridge-Field

  5. Review: Intervallic complementarity 2013 Eleanor Selfridge-Field

  6. Review: Intervallic complementarity in chords 2013 Eleanor Selfridge-Field

  7. Need for integer arithmetic • Binomial solutions: Brinkman, Böker-Heil • Required 3 params (pitch name, octave number, inflection) • Arbitrary mappings: C=10, D=20, E=30…. • Same intervals do not always produced same numbers (depends on endpoints: F-E = 10, Eb-D = 9) • Base-40 produces consistent arithmetical results (irrespective of endpoints and without binomials) 2013 Eleanor Selfridge-Field

  8. Physical instrument Cultural apparatus Letter names Base-7 (0 #s/bs) Octave numbers Base-12 (1#/b) Inflection names Base-21 (1#/b) Inflection names Base-40 (2#/b) Enharmonic-notation tiers 2013 Eleanor Selfridge-Field

  9. Third tier ## # - b bb Fourth tier ### ## # - b bb bbb Wider system: Enharmonic-notation tiers (7 x 5) + 5 C## / D / Ebb A# / Bb / Cbb D## / E / Fb 2013 Eleanor Selfridge-Field

  10. Base-40 Rule Simple rule: Where a whole step exists between two key names, a null token is used. 2013 Eleanor Selfridge-Field

  11. From Base-40 for enharmonic preservation MIDI to base-7 MIDI to base-12 MIDI to base-21 MIDI to base-40 Solution: Translate from symbolic code to MIDIPlus 2013 Eleanor Selfridge-Field

  12. What is MIDIPlus? • In MIDI file format, a binary implementation of base-40 • Replaces last 3 bits of velocity byte • Used to interpret key number 2013 Eleanor Selfridge-Field

  13. MIDIPlus in Printing Raw MIDI to Notation (Bach Prelude in E Minor, BWV 855 Translation from symbolic code (MuseData) to MIDIPlus to notation 2013 Eleanor Selfridge-Field

  14. Chords (interval complementarity) • Intervallic complementarity • Chord definitions 2013 Eleanor Selfridge-Field

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